Rayleigh-Taylor Instability in Confined Geometries

Tedi Godfrey
Advisor: Dr. Joshua Bostwick

Abstract

The Rayleigh-Taylor Instability (RTI) is seen when a heavy fluid is placed above a lighter fluid resulting in the instability of the interface into finger-like structures. When the fluids are confined by a small geometry, it is known that there is a critical depth in which the RTI can be suppressed. I use analytical techniques and numerical simulations to study this problem, as it depends upon the physical parameters. Numerical simulations of this multiphase flow are done using a combination of the multiphase solver in OpenFoam, an open-source computational fluid dynamics toolbox, and ParaView, which allows the results to be visualized. Our results show that an initially flat interface will become corrugated, resembling a wave, and develop the finger-like structures seen in RTI. The initial conditions of the simulation state that the interface between the two fluids is flat initially. A hydrodynamic stability analysis was done to derive a dispersion relation that relates the spatial wave number and temporal frequency of the wave using first principles. Dimensional analysis allows us to reduce the number of variables in the dispersion relation and isolate the effects that individual nondimensional parameters have on the dominant wavelength and critical depth.

Analytical Model

Governing Equations

  • Navier-Stokes Equation:

navier_stokes_equation

  • Incompressibility:

incompressibility

Assumptions

  1. Constant Density
  2. Inviscid – kinematic viscosity is zero
  3. Irrotational – curl of velocity vector field is zero

Stokes’ Theorem: vector field can be completely described by its scalar potential

stokes_theorem

Reduces to Bernoulli’s Theorem:

bernoulli_theorem

The two different fluids are represented by 1 and 2.

Velocity potentials independently satisfy Laplace’s Equation:

laplace_equation

And match at the interface:

match_at_interface

The interface between the more dense fluid, 1 and the less dense fluid, 2 is initially flat. As gravity accelerates fluid 1 into fluid 2, the interface becomes unstable, creating an exponentially growing perturbation described by the deviation.

Boundary Conditions

  1. Kinematic – vertical velocities of fluid match with interface
  2. Continuous Pressure
  3. Confined Geometry – free slip at walls (no friction between the walls and fluid)

Dispersion Relation

relates spatial wave number and temporal frequency of a wave.

  • Linearize equations – everything is small, drop second order terms
  • Ansatz – guess the form of the solution
  • Plug into Laplace’s equation
  • Solve for unknown terms with boundary conditions
  • Deviation:

deviation

  • Substitute, evaluate at z =0, solve for frequency
  • Dispersion Relation:

dispersion_relation

Solving this for the heavy over light configuration (rho 1 > rho 2) gives that the omega squared term is less than zero, indicating an unstable interface.

Now we seek to answer the question: what is the dominant spatial wavenumber of instability? We experiment with this using a numerical model.

Numerical Model

OpenFOAM v1912 – opensource computational fluid dynamics toolbox

  • interFoam – volume of fluid solver
  • finite volume method – approximates solutions
  • discretizes volume of each fluid into a finite number of volume elements
  • conservation of momentum – describes cell interaction

ParaView 5.8.0 – opensource visualization and data analysis software

 

Internal Mesh

We define the lateral length of the mesh to always be 4 times the depth. The mesh density was set to 80 x 320.

0.001 Scale Mesh

The heavier fluid is a glycerol-water mixture, and the lighter fluid is water.

Boundary Conditions

  1. velocity: no slip – zero velocity at walls
  2. pressure: zero gradient – no change in the magnitude from patch internal field to patch faces
  3. side walls: far-field/periodic – no end effects

Numerical values must be given to all physical parameters to get an approximate solution.

Parameters
Simulation

Results

There is a critical depth in which the Rayleigh-Taylor instability no longer occurs. This happens when a stabilizing term such as diffusion dominates the growth rate of the instability.  Using the previous mesh specifications, our simulation found this critical depth to be 1 mm by incrementally decreasing the depth until the instability no longer occurred.

Varying Parameters

The heavier fluid is an aqueous glycerol solution. Knowing an ideal range for the percent weight of each component of the mixture, and how it effects the dominant wavelength will help in future experimentation.

The dominant wavelength was found by summing the vertical pixel intensities, then taking the FFT (Fast Fourier transform).

The magnitude of the FFT is proportional to the Power Spectral Density, where the peak is the dominant wavelength.

We ran simulations for different percent mass of glycerol at various cell depths to measure the effects on the dominate wavelength.

Conclusions

We found that increasing the percent mass of glycerol, which increase the density and viscosity of the top layer, increases the dominant wavelength. Increasing the cell-depth also increases the dominant wave length.

Future Work

  • In-lab experimentation
  • Effects of other parameters and dominant wavelength
  • Refining mesh

References

Samar Alqatari. (2019). Reduced-Dimension Model for the Rayleigh-Taylor Instability in a Hele-Shaw Cell. MassachusettsInstitute of Technology.

C. P. Caulfield. (2017). Hydrodynamic Stability (Dexter Chua, Ed.).  https://dec41.user.srcf.net/notes/IIIM/hydrodynamicstabilitytrim.pdf

Jose Lorenzo. (2018). rayleighTaylorCase. https://github.com/joslorgom/rayleighTaylorCase

F. Moukalled, L. Mangani, M. Darwish. (2016). The Finite Volume Method in Computational Fluid Dynamics (Vol. 113). Springer.

The OpenFOAM Foundation. (2017). OpenFOAM v8 User Guide. https://cfd.direct/openfoam/user-guide

Solution Initialization Using codeStream. (2018). https ://www.cfd-china.com/assets/uploads/files/1535073081321-programmingofics.pdf